Integrand size = 14, antiderivative size = 165 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {4 (c+d x)}{3 b^2 d \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \]
-2/3*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d+2 /3*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d/exp(a/b)-2/ 3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^(3/2)-4/3*(d*x+ c)/b^2/d/(a+b*arccosh(d*x+c))^(1/2)
Time = 0.73 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\frac {e^{-\frac {a+b \text {arccosh}(c+d x)}{b}} \left (2 e^{\frac {2 a}{b}+\text {arccosh}(c+d x)} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )-2 \left (e^{a/b} \left (b e^{\text {arccosh}(c+d x)} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)+\left (1+e^{2 \text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+b e^{\text {arccosh}(c+d x)} \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )\right )}{3 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}} \]
(2*E^((2*a)/b + ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcC osh[c + d*x])*Gamma[1/2, a/b + ArcCosh[c + d*x]] - 2*(E^(a/b)*(b*E^ArcCosh [c + d*x]*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + (1 + E^(2*Arc Cosh[c + d*x]))*(a + b*ArcCosh[c + d*x])) + b*E^ArcCosh[c + d*x]*(-((a + b *ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)]))/( 3*b^2*d*E^((a + b*ArcCosh[c + d*x])/b)*(a + b*ArcCosh[c + d*x])^(3/2))
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6410, 6295, 6366, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 6410 |
\(\displaystyle \frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6295 |
\(\displaystyle \frac {\frac {2 \int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}}{d}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle \frac {\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)}{b}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}}{d}\) |
\(\Big \downarrow \) 6296 |
\(\displaystyle \frac {\frac {2 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {2 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}}{d}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}\right )}{3 b}}{d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-i \int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2}\right )}{3 b}}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}}{d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}}{d}\) |
((-2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*b*(a + b*ArcCosh[c + d*x])^( 3/2)) + (2*((-2*(c + d*x))/(b*Sqrt[a + b*ArcCosh[c + d*x]]) + ((2*I)*((I/2 )*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] - ((I /2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/E^(a/b))) /b^2))/(3*b))/d
3.2.90.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c /(b*(n + 1)) Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \frac {1}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]